Integrand size = 17, antiderivative size = 91 \[ \int \frac {1}{\sqrt {a+i a \text {csch}(c+d x)}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}}\right )}{\sqrt {a} d}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {2} \sqrt {a+i a \text {csch}(c+d x)}}\right )}{\sqrt {a} d} \]
2*arctanh(coth(d*x+c)*a^(1/2)/(a+I*a*csch(d*x+c))^(1/2))/d/a^(1/2)-arctanh (1/2*coth(d*x+c)*a^(1/2)*2^(1/2)/(a+I*a*csch(d*x+c))^(1/2))*2^(1/2)/d/a^(1 /2)
Time = 1.78 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.30 \[ \int \frac {1}{\sqrt {a+i a \text {csch}(c+d x)}} \, dx=\frac {\sqrt {a} \left (2 \arctan \left (\frac {\sqrt {i a (i+\text {csch}(c+d x))}}{\sqrt {a}}\right )-\sqrt {2} \arctan \left (\frac {\sqrt {i a (i+\text {csch}(c+d x))}}{\sqrt {2} \sqrt {a}}\right )\right ) \coth (c+d x)}{d \sqrt {i a (i+\text {csch}(c+d x))} \sqrt {a+i a \text {csch}(c+d x)}} \]
(Sqrt[a]*(2*ArcTan[Sqrt[I*a*(I + Csch[c + d*x])]/Sqrt[a]] - Sqrt[2]*ArcTan [Sqrt[I*a*(I + Csch[c + d*x])]/(Sqrt[2]*Sqrt[a])])*Coth[c + d*x])/(d*Sqrt[ I*a*(I + Csch[c + d*x])]*Sqrt[a + I*a*Csch[c + d*x]])
Time = 0.40 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {3042, 4263, 26, 3042, 26, 4261, 216, 4282, 216}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{\sqrt {a+i a \text {csch}(c+d x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sqrt {a-a \csc (i c+i d x)}}dx\) |
\(\Big \downarrow \) 4263 |
\(\displaystyle \int -\frac {i \text {csch}(c+d x)}{\sqrt {i \text {csch}(c+d x) a+a}}dx+\frac {\int \sqrt {i \text {csch}(c+d x) a+a}dx}{a}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {\int \sqrt {i \text {csch}(c+d x) a+a}dx}{a}-i \int \frac {\text {csch}(c+d x)}{\sqrt {i \text {csch}(c+d x) a+a}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \sqrt {a-a \csc (i c+i d x)}dx}{a}-i \int \frac {i \csc (i c+i d x)}{\sqrt {a-a \csc (i c+i d x)}}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \int \frac {\csc (i c+i d x)}{\sqrt {a-a \csc (i c+i d x)}}dx+\frac {\int \sqrt {a-a \csc (i c+i d x)}dx}{a}\) |
\(\Big \downarrow \) 4261 |
\(\displaystyle \int \frac {\csc (i c+i d x)}{\sqrt {a-a \csc (i c+i d x)}}dx-\frac {2 i \int \frac {1}{a-\frac {a^2 \coth ^2(c+d x)}{i \text {csch}(c+d x) a+a}}d\frac {i a \coth (c+d x)}{\sqrt {i \text {csch}(c+d x) a+a}}}{d}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \int \frac {\csc (i c+i d x)}{\sqrt {a-a \csc (i c+i d x)}}dx+\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}}\right )}{\sqrt {a} d}\) |
\(\Big \downarrow \) 4282 |
\(\displaystyle \frac {2 i \int \frac {1}{2 a-\frac {a^2 \coth ^2(c+d x)}{i \text {csch}(c+d x) a+a}}d\frac {i a \coth (c+d x)}{\sqrt {i \text {csch}(c+d x) a+a}}}{d}+\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}}\right )}{\sqrt {a} d}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {2 \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}}\right )}{\sqrt {a} d}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {2} \sqrt {a+i a \text {csch}(c+d x)}}\right )}{\sqrt {a} d}\) |
(2*ArcTanh[(Sqrt[a]*Coth[c + d*x])/Sqrt[a + I*a*Csch[c + d*x]]])/(Sqrt[a]* d) - (Sqrt[2]*ArcTanh[(Sqrt[a]*Coth[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Csch[c + d*x]])])/(Sqrt[a]*d)
3.1.54.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(b/d) Subst[Int[1/(a + x^2), x], x, b*(Cot[c + d*x]/Sqrt[a + b*Csc[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Int[1/Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[1/a I nt[Sqrt[a + b*Csc[c + d*x]], x], x] - Simp[b/a Int[Csc[c + d*x]/Sqrt[a + b*Csc[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S ymbol] :> Simp[-2/f Subst[Int[1/(2*a + x^2), x], x, b*(Cot[e + f*x]/Sqrt[ a + b*Csc[e + f*x]])], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]
\[\int \frac {1}{\sqrt {a +i a \,\operatorname {csch}\left (d x +c \right )}}d x\]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 551 vs. \(2 (72) = 144\).
Time = 0.29 (sec) , antiderivative size = 551, normalized size of antiderivative = 6.05 \[ \int \frac {1}{\sqrt {a+i a \text {csch}(c+d x)}} \, dx=-\frac {1}{2} \, \sqrt {2} \sqrt {\frac {1}{a d^{2}}} \log \left (2 \, {\left (\sqrt {2} {\left (a d e^{\left (2 \, d x + 2 \, c\right )} - a d\right )} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} \sqrt {\frac {1}{a d^{2}}} + a e^{\left (d x + c\right )} - i \, a\right )} e^{\left (-d x - c\right )}\right ) + \frac {1}{2} \, \sqrt {2} \sqrt {\frac {1}{a d^{2}}} \log \left (-2 \, {\left (\sqrt {2} {\left (a d e^{\left (2 \, d x + 2 \, c\right )} - a d\right )} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} \sqrt {\frac {1}{a d^{2}}} - a e^{\left (d x + c\right )} + i \, a\right )} e^{\left (-d x - c\right )}\right ) + \frac {1}{2} \, \sqrt {\frac {1}{a d^{2}}} \log \left (\frac {2 \, {\left ({\left (d e^{\left (2 \, d x + 2 \, c\right )} - d\right )} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} \sqrt {\frac {1}{a d^{2}}} + e^{\left (d x + c\right )} + i\right )} e^{\left (-d x - c\right )}}{d}\right ) - \frac {1}{2} \, \sqrt {\frac {1}{a d^{2}}} \log \left (-\frac {2 \, {\left ({\left (d e^{\left (2 \, d x + 2 \, c\right )} - d\right )} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} \sqrt {\frac {1}{a d^{2}}} - e^{\left (d x + c\right )} - i\right )} e^{\left (-d x - c\right )}}{d}\right ) + \frac {1}{2} \, \sqrt {\frac {1}{a d^{2}}} \log \left (\frac {2 \, {\left ({\left (a d e^{\left (2 \, d x + 2 \, c\right )} - i \, a d e^{\left (d x + c\right )} - 2 \, a d\right )} \sqrt {\frac {1}{a d^{2}}} + \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} {\left (e^{\left (3 \, d x + 3 \, c\right )} - 2 i \, e^{\left (2 \, d x + 2 \, c\right )} - e^{\left (d x + c\right )} + 2 i\right )}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right ) - \frac {1}{2} \, \sqrt {\frac {1}{a d^{2}}} \log \left (-\frac {2 \, {\left ({\left (a d e^{\left (2 \, d x + 2 \, c\right )} - i \, a d e^{\left (d x + c\right )} - 2 \, a d\right )} \sqrt {\frac {1}{a d^{2}}} - \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} {\left (e^{\left (3 \, d x + 3 \, c\right )} - 2 i \, e^{\left (2 \, d x + 2 \, c\right )} - e^{\left (d x + c\right )} + 2 i\right )}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right ) \]
-1/2*sqrt(2)*sqrt(1/(a*d^2))*log(2*(sqrt(2)*(a*d*e^(2*d*x + 2*c) - a*d)*sq rt(a/(e^(2*d*x + 2*c) - 1))*sqrt(1/(a*d^2)) + a*e^(d*x + c) - I*a)*e^(-d*x - c)) + 1/2*sqrt(2)*sqrt(1/(a*d^2))*log(-2*(sqrt(2)*(a*d*e^(2*d*x + 2*c) - a*d)*sqrt(a/(e^(2*d*x + 2*c) - 1))*sqrt(1/(a*d^2)) - a*e^(d*x + c) + I*a )*e^(-d*x - c)) + 1/2*sqrt(1/(a*d^2))*log(2*((d*e^(2*d*x + 2*c) - d)*sqrt( a/(e^(2*d*x + 2*c) - 1))*sqrt(1/(a*d^2)) + e^(d*x + c) + I)*e^(-d*x - c)/d ) - 1/2*sqrt(1/(a*d^2))*log(-2*((d*e^(2*d*x + 2*c) - d)*sqrt(a/(e^(2*d*x + 2*c) - 1))*sqrt(1/(a*d^2)) - e^(d*x + c) - I)*e^(-d*x - c)/d) + 1/2*sqrt( 1/(a*d^2))*log(2*((a*d*e^(2*d*x + 2*c) - I*a*d*e^(d*x + c) - 2*a*d)*sqrt(1 /(a*d^2)) + sqrt(a/(e^(2*d*x + 2*c) - 1))*(e^(3*d*x + 3*c) - 2*I*e^(2*d*x + 2*c) - e^(d*x + c) + 2*I))*e^(-2*d*x - 2*c)/d) - 1/2*sqrt(1/(a*d^2))*log (-2*((a*d*e^(2*d*x + 2*c) - I*a*d*e^(d*x + c) - 2*a*d)*sqrt(1/(a*d^2)) - s qrt(a/(e^(2*d*x + 2*c) - 1))*(e^(3*d*x + 3*c) - 2*I*e^(2*d*x + 2*c) - e^(d *x + c) + 2*I))*e^(-2*d*x - 2*c)/d)
\[ \int \frac {1}{\sqrt {a+i a \text {csch}(c+d x)}} \, dx=\int \frac {1}{\sqrt {i a \operatorname {csch}{\left (c + d x \right )} + a}}\, dx \]
\[ \int \frac {1}{\sqrt {a+i a \text {csch}(c+d x)}} \, dx=\int { \frac {1}{\sqrt {i \, a \operatorname {csch}\left (d x + c\right ) + a}} \,d x } \]
\[ \int \frac {1}{\sqrt {a+i a \text {csch}(c+d x)}} \, dx=\int { \frac {1}{\sqrt {i \, a \operatorname {csch}\left (d x + c\right ) + a}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt {a+i a \text {csch}(c+d x)}} \, dx=\int \frac {1}{\sqrt {a+\frac {a\,1{}\mathrm {i}}{\mathrm {sinh}\left (c+d\,x\right )}}} \,d x \]